Robert Mancini was as good as his word when he talked about tackling Manchester United's monumental ascendancy. "We have five games to recover six or seven points," the Manchester City manager declared on Friday, and in the performance of his team, and Samir Nasri in particular, here was irrefutable evidence that the deposed champions have not slipped into a mood of resigned and gloomy indifference.

The game ended with one of those moments of misery which have stalked Joe Hart from Southampton to Sunderland this season, when he allowed Andy Carroll's strong but by no means unstoppable shot to slip through his legs. Yet the pained look on his face as he left the field suggested that he had been feeling the effects of an injury which had immobilised him as the ball rolled over the line.

Hart had suffered a back spasm which left him prone on the pitch, with a stretcher waiting, five minutes before conceding that injury time goal, when he leapt and twisted to claw away a header from Carroll.

"It's a 24-hour thing," David Platt, City's assistant manager, said of the injury. "Most keepers will have an issue like that because they are throwing themselves about left, right and centre during the week. He will be a bit sore tomorrow but it doesn't matter."

If the goal had not come so late, it might have created some suspense, but the course of the afternoon was never really in doubt. Sam Allardyce's side lacked ambition and, according to the manager, showed City too much respect for the first 45 minutes. Their immediate strategy was to load the rear quarters and intimidate David Silva.

Nasri was the shining light, though Platt rejected suggestions that the Frenchman's excellence in the past few weeks was a response to "Roberto giving him a coating", as he put it.

Platt denied there had been any "coating" at all – a questionable assertion – but the 25-year-old has been wearing a different face in the last few weeks. There were half-a-dozen excellent moments from him – a sharp shot on the half-hour, forcing Jussi Jaaskelainen into the first of a smart double save, and an elegant run through the visiting defence to feed Carlos Tevez.

It was, however, his quick, precision sequence of one-twos with Yaya Touré and Silva which proved decisive, paving the way for Sergio Aguero to slot in City's opening goal at the near post.

Allardyce's players showed more ambition after the break, though that created space for City's counter-attacking football, of which Tevez – another player defying assertions about his best City performances belonging in the past – was the fulcrum. Tevez's gorgeous reverse pass set free Nasri for a shot saved at point-blank range by the excellent Jaaskelainen.

City could have put the game beyond doubt after 70 minutes had Nasri converted the gilt-edged chance presented by the dangerous Aguero's cross from the right, at the end of another searing counter-attack. But the midfielder tried to guide the ball in with his right instep, when his left foot looked like the natural weapon.

City were nearly punished for this profligacy when, with Hart on the ground and helpless after Matt Jarvis's shot had been blocked, James Collins rifled a follow-up effort wide on 82 minutes. "Had James hit the target, Joe Hart was on the floor, stranded," Allardyce reflected ruefully.

Yaya Touré immediately and emphatically finished the job, taking a ball from Nasri and powering an accutate shot into the top corner of Jaaskelainen's goal from just outside the 18-yard box.

And only then did Carroll fire into life. Allardyce again confirmed last night that he "would like to" sign the striker permanently from Liverpool. "We'll find out at the end of the season. He's scored again hasn't he? Six in nine now…"

Mancini professed himself slightly unhappy. "We want to be too beautiful, maybe. It is enough just to score. It's [been] the same [all] season." But this was enough to show that City want to maintain a reputation, even if they have relinquished a title.